In this presentation we describe the Fourier coefficients of a few common signals. We illustrate how irregularities (discontinuities) in the signal correspond to a slow decay of weights on larger harmonics.

This presentation was made with

• ioslides by RStudio
• python's numpy for fast Fourier transforms and inverses
• python's version of plotly
• the reticulate package for executing python in RStudio

import json
import plotly
import plotly.graph_objs as go

import numpy as np
from numpy.fft import fft, ifft
import math
eps = 1e-3

def plot_signal_and_fft(signal, name, N=20, asFigure=False):
    f = fft(signal)
    N = min(N, len(signal) / 2)
    x = np.arange(-N, N)
    y = np.concatenate((f[-N:], f[:N]))
    f[N:len(f) - N] = 0
    signal_reconstructed = ifft(f).real
    data = [go.Scatter(x=np.arange(-len(signal), len(signal)),
                       y=np.concatenate((signal, signal)),
                       name='Discrete signal'),
            go.Scatter(x=np.arange(-len(signal), len(signal)),
                       y=np.concatenate(
                           (signal_reconstructed, signal_reconstructed)),
                       name='Signal reconstructed from truncated FFT'),
            go.Bar(x=x, y=y.real,
                   name='Fourier transform (real part)',
                   xaxis='x2', yaxis='y2'),
            go.Bar(x=x, y=y.imag,
                   name='Fourier transform (imaginary part)',
                   xaxis='x2', yaxis='y2')]
    layout = go.Layout(title='Fourier transform for {}'.format(name),
                       barmode='stack',
                       xaxis=dict(domain=[0, .45]),
                       xaxis2=dict(domain=[.55, 1]),
                       yaxis=dict(title='amplitude'),
                       yaxis2=dict(anchor='x2'),
                       legend=dict(orientation='h'))
    fig = dict(data=data, layout=layout)
    return json.dumps(fig, cls=plotly.utils.PlotlyJSONEncoder)

The Fourier of a sinusoidal is 100% imaginary, and has positive value on the main frequency only.

x = np.arange(0, 2 * math.pi, eps)
fig = plot_signal_and_fft(np.sin(x), 'sinus')

The Fourier of a cosinus is 100% real, and has positive value on the main frequency only.

fig = plot_signal_and_fft(np.cos(x), 'cos')

The triangle wave has a few harmonics, but they strength decrease rather quickly. The signal reconstructed from 20 harmonics matches very well the original signal.

x = np.arange(-1, 1, eps)
y = np.abs(x) - .5
fig = plot_signal_and_fft(y, 'triangle')

The signal is discontinuous and as such, the Fourier coefficient decrease slowly. A truncated Fourier series corresponds to a continuous signal that matches only approximately the original signal.

y = np.concatenate((np.ones(100), -np.ones(100)))
fig = plot_signal_and_fft(y, 'square')

This signal is also discontinuous and again, the Fourier coefficient decrease slowly. A truncated Fourier series corresponds to a continuous signal that matches only approximately the original signal.

x = np.arange(0, 1, eps)
y = np.abs(x) - .5
fig = plot_signal_and_fft(y, 'sawtooth')